A263647 Numbers n such that 2^n-1 and 3^n-1 are coprime.

1, 2, 3, 5, 7, 9, 13, 14, 15, 17, 19, 21, 25, 26, 27, 29, 31, 34, 37, 38, 39, 41, 45, 47, 49, 51, 53, 57, 59, 61, 62, 63, 65, 67, 71, 73, 74, 79, 81, 85, 87, 89, 91, 93, 94, 97, 98, 101, 103, 107, 109, 111, 113, 118, 122, 123, 125, 127, 133, 134, 135, 137, 139, 141, 142, 145, 147, 149, 151, 153, 157, 158, 159, 163, 167, 169, 171

Offset

1,2

Comments

n such that there is no k for which both A014664(k) and A062117(k) divide n.

If n is in the sequence, then so is every divisor of n.

1 and 2 are the only members that are in A006093.

Conjectured to be infinite: see the Ailon and Rudnick paper.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

N. Ailon and Z. Rudnick, Torsion points on curves and common divisors of a^k - 1 and b^k - 1, Acta Arith. 113 (2004), 31-38. Also arXiv:math/0202102 [math.NT], 2002.

Example

gcd(2^1-1, 3^1-1) = gcd(1,2) = 1, so a(1) = 1.
gcd(2^2-1, 3^2-1) = gcd(3,8) = 1, so a(2) = 2.
gcd(2^4-1, 3^4-1) = gcd(15,80) = 5, so 4 is not in the sequence.

Maple

select(n -> igcd(2^n-1, 3^n-1)=1, [$1..1000]);

Mathematica

Select[Range[200], GCD[2^# - 1, 3^# - 1] == 1 &] (* Vincenzo Librandi, May 01 2016 *)

Prog

(Magma) [n: n in [1..200] | Gcd(2^n-1, 3^n-1) eq 1]; // Vincenzo Librandi, May 01 2016

Crossrefs

Cf. A000225, A006093, A024023, A014664, A062117.

Sequence in context: A333573 A361852 A032459 * A028870 A338356 A057886

Adjacent sequences: A263644 A263645 A263646 * A263648 A263649 A263650

Keyword

nonn

Author

Robert Israel, Oct 22 2015

Status

approved